Unbiased `walk-on-spheres' Monte Carlo methods for the fractional Laplacian
This provides a novel, unbiased Monte Carlo algorithm for fractional PDEs, which are important in anomalous diffusion and nonlocal phenomena, but the work is incremental as it adapts existing walk-on-spheres ideas to a different process.
The authors developed an unbiased walk-on-spheres Monte Carlo method for solving fractional Laplacian boundary-value problems, using isotropic alpha-stable processes that exit spheres by jumps, ensuring finite termination. The method is unbiased and handles disconnected domains.
We consider Monte Carlo methods for simulating solutions to the analogue of the Dirichlet boundary-value problem in which the Laplacian is replaced by the fractional Laplacian and boundary conditions are replaced by conditions on the exterior of the domain. Specifically, we consider the analogue of the so-called `walk-on-spheres` algorithm. In the diffusive setting, this entails sampling the path of Brownian motion as it uniformly exits a sequence of spheres maximally inscribed in the domain. As this algorithm would otherwise never end, it is truncated when the `walk-on-spheres` comes within epsilon > 0 of the boundary. In the setting of the fractional Laplacian, the role of Brownian motion is replaced by an isotropic alpha-stable process with alpha in (0, 2). A significant difference to the Brownian setting is that the stable processes will exit spheres by a jump rather than hitting their boundary. This difference ensures that disconnected domains may be considered and that, unlike the diffusive setting, the algorithm ends after an almost surely finite number of steps.